Introduction to Open Data Science - Course Project

My thoughts after the first lecture

The course seems interesting but quite hard. Let’s see. I’m hoping to learn how to use GitHub and R Markdown and better my skills with R. I heard about the course through an email send by my doctoral school.

https://github.com/SarKin-bit/IODS-project

# This is a so-called "R chunk" where you can write R code.

date()
## [1] "Fri Nov 27 00:09:01 2020"

The text continues here…


Exercise 2 Regression and model validation

learning2014<- read.delim("learning2014.txt")

dim(learning2014)
## [1] 166   7
str(learning2014)
## 'data.frame':    166 obs. of  7 variables:
##  $ gender  : chr  "F" "M" "F" "M" ...
##  $ Age     : int  53 55 49 53 49 38 50 37 37 42 ...
##  $ Attitude: int  37 31 25 35 37 38 35 29 38 21 ...
##  $ deep    : num  3.58 2.92 3.5 3.5 3.67 ...
##  $ stra    : num  3.38 2.75 3.62 3.12 3.62 ...
##  $ surf    : num  2.58 3.17 2.25 2.25 2.83 ...
##  $ Points  : int  25 12 24 10 22 21 21 31 24 26 ...

There are 166 observations (rows) and 7 variables (columns). Columns are called gender, age, attitude, deep, stra, surf and points. All columns include data in numerical form, except column “gender”, which includes only characters.

Graphical overview of the data: Initialize plot with data and aesthetic mapping.

library(ggplot2)
p1 <- ggplot(learning2014, aes(x = Attitude, y = Points, col = gender))

Define the visualization type (points).

p2 <- p1 + geom_point()

Draw the plot.

p2

Add a regression line.

p3 <- p2 + geom_smooth(method = "lm")
p3
## `geom_smooth()` using formula 'y ~ x'

Summaries of the variables in the data.

summary(learning2014)
##     gender               Age           Attitude          deep      
##  Length:166         Min.   :17.00   Min.   :14.00   Min.   :1.583  
##  Class :character   1st Qu.:21.00   1st Qu.:26.00   1st Qu.:3.333  
##  Mode  :character   Median :22.00   Median :32.00   Median :3.667  
##                     Mean   :25.51   Mean   :31.43   Mean   :3.680  
##                     3rd Qu.:27.00   3rd Qu.:37.00   3rd Qu.:4.083  
##                     Max.   :55.00   Max.   :50.00   Max.   :4.917  
##       stra            surf           Points     
##  Min.   :1.250   Min.   :1.583   Min.   : 7.00  
##  1st Qu.:2.625   1st Qu.:2.417   1st Qu.:19.00  
##  Median :3.188   Median :2.833   Median :23.00  
##  Mean   :3.121   Mean   :2.787   Mean   :22.72  
##  3rd Qu.:3.625   3rd Qu.:3.167   3rd Qu.:27.75  
##  Max.   :5.000   Max.   :4.333   Max.   :33.00

There seems to be a somewhat positive correlation here but distributions of the variables are scattered quite a bit and some are quite far from the regression line, meaning that the fit of the model is not perfect.

Setting three variables as explanatory variables and fitting a regression model where exam points is the (dependent) variable:

Create an plot matrix with ggpairs().

library(GGally)
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
library(ggplot2)
ggpairs(learning2014, lower = list(combo = wrap("facethist", bins = 20)))

Create a regression model with multiple explanatory variables.

my_model2 <- lm(Points ~ Attitude + stra + deep, data = learning2014)

Print out a summary of the model.

summary(my_model2)
## 
## Call:
## lm(formula = Points ~ Attitude + stra + deep, data = learning2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.5239  -3.4276   0.5474   3.8220  11.5112 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.39145    3.40775   3.343  0.00103 ** 
## Attitude     0.35254    0.05683   6.203 4.44e-09 ***
## stra         0.96208    0.53668   1.793  0.07489 .  
## deep        -0.74920    0.75066  -0.998  0.31974    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.289 on 162 degrees of freedom
## Multiple R-squared:  0.2097, Adjusted R-squared:  0.195 
## F-statistic: 14.33 on 3 and 162 DF,  p-value: 2.521e-08

Relationship with the explanatory variable “Attitude” and response variable “Points” is statistically significant (p<0.05). Relationship with the explanatory variables “stra” and “deep” are not statistically significant with response variable “Points” (p>0.05).

Redo the model without the variables that had no statistical significance. I did a simple regression as two of the explanatory variables had no significant effect.

A scatter plot of points versus attitude.

qplot(Attitude, Points, data = learning2014) + geom_smooth(method = "lm")
## `geom_smooth()` using formula 'y ~ x'

Fit a linear model.

my_model3 <- lm(Points ~ Attitude, data = learning2014)
summary(my_model3)
## 
## Call:
## lm(formula = Points ~ Attitude, data = learning2014)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.9763  -3.2119   0.4339   4.1534  10.6645 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.63715    1.83035   6.358 1.95e-09 ***
## Attitude     0.35255    0.05674   6.214 4.12e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared:  0.1906, Adjusted R-squared:  0.1856 
## F-statistic: 38.61 on 1 and 164 DF,  p-value: 4.119e-09

Relationship between the target variable (Points) and the explanatory variable (Attitude) is statistically significant (p<0.05). Multiple R squared is 0.1906. It is quite low (compared to 1), which means that the data is not that close to the fitted regression line.This indicates that the model explains only some of the variability of the response data around its mean. The model does not seem to fit the data very well.

Draw diagnostic plots Residuals vs Fitted values, Normal QQ-plot and Residuals vs Leverage.

par(mfrow = c(2,2))
plot(my_model2, which = c(1,2,5))

Assumptions of the model are of that of linear regression:Linear relationship, multivariate normality, no or little multicollinearity, no auto-correlation and homoscedasticity.

In the Residuals vs Fitted values the red line is not perfectly flat, which indicates that there is discernible non-linear trend to the residuals. The residuals do not appear to be equally distributed across the entire range of the fitted values.

In the Normal QQ-plot the we can see thet the residuals are not normally distributed as the residual devide from the diagonal line.

In the Residuals vs Leverage there are no cases beyond the Cook’s distance lines which means that there is not any influential cases, i.e. there are no influential outliers.

date()
## [1] "Fri Nov 27 00:09:08 2020"

Here we go again…


Exercise 3 Logistic regression

alc<-read.table(file = "alc.txt", sep="\t", header=TRUE)

dim(alc)
## [1] 382  35
str(alc)
## 'data.frame':    382 obs. of  35 variables:
##  $ school    : chr  "GP" "GP" "GP" "GP" ...
##  $ sex       : chr  "F" "F" "F" "F" ...
##  $ age       : int  18 17 15 15 16 16 16 17 15 15 ...
##  $ address   : chr  "U" "U" "U" "U" ...
##  $ famsize   : chr  "GT3" "GT3" "LE3" "GT3" ...
##  $ Pstatus   : chr  "A" "T" "T" "T" ...
##  $ Medu      : int  4 1 1 4 3 4 2 4 3 3 ...
##  $ Fedu      : int  4 1 1 2 3 3 2 4 2 4 ...
##  $ Mjob      : chr  "at_home" "at_home" "at_home" "health" ...
##  $ Fjob      : chr  "teacher" "other" "other" "services" ...
##  $ reason    : chr  "course" "course" "other" "home" ...
##  $ nursery   : chr  "yes" "no" "yes" "yes" ...
##  $ internet  : chr  "no" "yes" "yes" "yes" ...
##  $ guardian  : chr  "mother" "father" "mother" "mother" ...
##  $ traveltime: int  2 1 1 1 1 1 1 2 1 1 ...
##  $ studytime : int  2 2 2 3 2 2 2 2 2 2 ...
##  $ failures  : int  0 0 2 0 0 0 0 0 0 0 ...
##  $ schoolsup : chr  "yes" "no" "yes" "no" ...
##  $ famsup    : chr  "no" "yes" "no" "yes" ...
##  $ paid      : chr  "no" "no" "yes" "yes" ...
##  $ activities: chr  "no" "no" "no" "yes" ...
##  $ higher    : chr  "yes" "yes" "yes" "yes" ...
##  $ romantic  : chr  "no" "no" "no" "yes" ...
##  $ famrel    : int  4 5 4 3 4 5 4 4 4 5 ...
##  $ freetime  : int  3 3 3 2 3 4 4 1 2 5 ...
##  $ goout     : int  4 3 2 2 2 2 4 4 2 1 ...
##  $ Dalc      : int  1 1 2 1 1 1 1 1 1 1 ...
##  $ Walc      : int  1 1 3 1 2 2 1 1 1 1 ...
##  $ health    : int  3 3 3 5 5 5 3 1 1 5 ...
##  $ absences  : int  5 3 8 1 2 8 0 4 0 0 ...
##  $ G1        : int  2 7 10 14 8 14 12 8 16 13 ...
##  $ G2        : int  8 8 10 14 12 14 12 9 17 14 ...
##  $ G3        : int  8 8 11 14 12 14 12 10 18 14 ...
##  $ alc_use   : num  1 1 2.5 1 1.5 1.5 1 1 1 1 ...
##  $ high_use  : logi  FALSE FALSE TRUE FALSE FALSE FALSE ...

Access the tidyverse libraries tidyr, dplyr, ggplot2.

library(tidyr); library(dplyr); library(ggplot2)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

Glimpse at the alc data.

glimpse(alc) 
## Rows: 382
## Columns: 35
## $ school     <chr> "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP"…
## $ sex        <chr> "F", "F", "F", "F", "F", "M", "M", "F", "M", "M", "F", "F"…
## $ age        <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15, 15, 15…
## $ address    <chr> "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U"…
## $ famsize    <chr> "GT3", "GT3", "LE3", "GT3", "GT3", "LE3", "LE3", "GT3", "L…
## $ Pstatus    <chr> "A", "T", "T", "T", "T", "T", "T", "A", "A", "T", "T", "T"…
## $ Medu       <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, 3, 3, 4…
## $ Fedu       <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, 3, 2, 3…
## $ Mjob       <chr> "at_home", "at_home", "at_home", "health", "other", "servi…
## $ Fjob       <chr> "teacher", "other", "other", "services", "other", "other",…
## $ reason     <chr> "course", "course", "other", "home", "home", "reputation",…
## $ nursery    <chr> "yes", "no", "yes", "yes", "yes", "yes", "yes", "yes", "ye…
## $ internet   <chr> "no", "yes", "yes", "yes", "no", "yes", "yes", "no", "yes"…
## $ guardian   <chr> "mother", "father", "mother", "mother", "father", "mother"…
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1…
## $ studytime  <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, 2, 1, 1…
## $ failures   <int> 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0…
## $ schoolsup  <chr> "yes", "no", "yes", "no", "no", "no", "no", "yes", "no", "…
## $ famsup     <chr> "no", "yes", "no", "yes", "yes", "yes", "no", "yes", "yes"…
## $ paid       <chr> "no", "no", "yes", "yes", "yes", "yes", "no", "no", "yes",…
## $ activities <chr> "no", "no", "no", "yes", "no", "yes", "no", "no", "no", "y…
## $ higher     <chr> "yes", "yes", "yes", "yes", "yes", "yes", "yes", "yes", "y…
## $ romantic   <chr> "no", "no", "no", "yes", "no", "no", "no", "no", "no", "no…
## $ famrel     <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, 5, 5, 3…
## $ freetime   <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, 3, 5, 1…
## $ goout      <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, 2, 5, 3…
## $ Dalc       <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1…
## $ Walc       <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1, 4, 3…
## $ health     <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, 4, 5, 5…
## $ absences   <int> 5, 3, 8, 1, 2, 8, 0, 4, 0, 0, 1, 2, 1, 1, 0, 5, 8, 3, 9, 5…
## $ G1         <int> 2, 7, 10, 14, 8, 14, 12, 8, 16, 13, 12, 10, 13, 11, 14, 16…
## $ G2         <int> 8, 8, 10, 14, 12, 14, 12, 9, 17, 14, 11, 12, 14, 11, 15, 1…
## $ G3         <int> 8, 8, 11, 14, 12, 14, 12, 10, 18, 14, 12, 12, 13, 12, 16, …
## $ alc_use    <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1.5, 1.0…
## $ high_use   <lgl> FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE, FAL…

Use gather() to gather columns into key-value pairs and then glimpse() at the resulting data.

gather(alc) %>% glimpse
## Rows: 13,370
## Columns: 2
## $ key   <chr> "school", "school", "school", "school", "school", "school", "sc…
## $ value <chr> "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP…

Draw a bar plot of each variable.

gather(alc) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar()

There are 382 observations (rows) and 35 variables (columns). Variables in the data are for example school, sex, age, adress, family size, family educational support, free time after school and internet access at home. For more detailed explanation of the variables, please visit: https://archive.ics.uci.edu/ml/datasets/Student+Performance

Chosen interesting variables: student’s grade, school, home address type and student’s health.

Hypothesis 1: There is a relationship between alcohol use and student’s grade. Hypothesis 2: There is a relationship between alcohol use and student’s absences. Hypothesis 3: There is a relationship between alcohol use and student’s health. Hypothesis 4: There is a relationship between alcohol use and student’s quality of family relationships.

Hypothesis 1: There is a relationship between alcohol use and student’s grade.

attach(alc)
table(G3,high_use,sex)
## , , sex = F
## 
##     high_use
## G3   FALSE TRUE
##   0      0    0
##   2      0    0
##   3      0    0
##   4      3    0
##   5      3    0
##   6     11    2
##   7      3    1
##   8     14    3
##   9      2    1
##   10    25    6
##   11    10    6
##   12    33   10
##   13     8    3
##   14    15    3
##   15    10    1
##   16    13    4
##   17     2    2
##   18     4    0
## 
## , , sex = M
## 
##     high_use
## G3   FALSE TRUE
##   0      1    1
##   2      0    1
##   3      1    0
##   4      3    1
##   5      1    2
##   6      4    5
##   7      1    1
##   8      5    5
##   9      1    3
##   10    13   20
##   11     6    4
##   12    21   17
##   13    11    4
##   14    16    4
##   15     5    0
##   16    17    4
##   17     3    0
##   18     3    0
library(ggplot2)

Initialise a plot of high_use and G3

g1 <- ggplot(alc, aes(x = high_use, y = G3, col = sex))

Define the plot as a boxplot and draw it.

g1 + geom_boxplot() + ylab("grade")

Initialise a plot of high use and sex.

g2<-ggplot(data = alc, aes(x = high_use, fill = sex))

Define the plot as a bar lot and draw it.

g2 + geom_bar()+facet_wrap("sex")

Male students who have high usage of alcohol (use a lot of alcohol) have lower grades than men who do not use a lot of alcohol. For female students, the high use of alcohol does not affect the grade.

Hypothesis 2: There is a relationship between alcohol use and student’s absences.

attach(alc)
## The following objects are masked from alc (pos = 3):
## 
##     absences, activities, address, age, alc_use, Dalc, failures,
##     famrel, famsize, famsup, Fedu, Fjob, freetime, G1, G2, G3, goout,
##     guardian, health, high_use, higher, internet, Medu, Mjob, nursery,
##     paid, Pstatus, reason, romantic, school, schoolsup, sex, studytime,
##     traveltime, Walc
table(G3,absences,sex)
## , , sex = F
## 
##     absences
## G3    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 16 17 18 19 20 21 26 27 29 44
##   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   2   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   3   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   4   1  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   5   0  0  2  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   6   5  3  0  1  1  2  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   7   0  0  0  1  0  0  0  0  0  2  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0
##   8   1  2  4  1  0  2  2  1  0  0  1  0  0  1  0  0  0  0  0  0  2  0  0  0  0
##   9   0  0  2  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   10  5  4  4  1  4  1  1  1  1  1  2  0  0  0  2  1  0  0  0  2  0  0  0  0  0
##   11  2  2  1  2  1  3  0  0  2  0  0  0  1  0  0  0  0  0  0  0  0  0  0  1  1
##   12  5  5 14  5  2  1  2  1  5  2  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0
##   13  0  2  2  2  1  0  1  2  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   14  2  4  0  3  3  2  0  1  1  0  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0
##   15  3  0  4  1  1  0  0  1  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0
##   16  5  1  3  0  1  2  3  0  0  1  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0
##   17  3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0
##   18  2  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##     absences
## G3   45
##   0   0
##   2   0
##   3   0
##   4   0
##   5   0
##   6   0
##   7   0
##   8   0
##   9   0
##   10  1
##   11  0
##   12  0
##   13  0
##   14  0
##   15  0
##   16  0
##   17  0
##   18  0
## 
## , , sex = M
## 
##     absences
## G3    0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 16 17 18 19 20 21 26 27 29 44
##   0   2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   2   0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   3   0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0
##   4   0  0  1  0  1  0  0  0  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   5   1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   6   1  1  1  3  0  0  0  0  0  1  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0
##   7   0  0  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   8   1  1  0  0  4  0  0  0  1  0  1  0  2  0  0  0  0  0  0  0  0  0  0  0  0
##   9   0  1  0  0  1  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  1  0  0  0
##   10  4  4  5  2  3  4  2  1  2  2  0  0  1  0  2  0  1  0  0  0  0  0  0  0  0
##   11  1  2  0  1  1  2  0  1  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0
##   12  8  8  4  3  2  1  1  2  2  3  1  1  0  0  1  0  0  1  0  0  0  0  0  0  0
##   13  2  2  2  5  3  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0
##   14  4  2  3  4  2  0  3  0  1  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0
##   15  0  1  0  1  2  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   16  5  4  3  1  2  1  2  0  2  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   17  1  1  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   18  1  0  0  1  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##     absences
## G3   45
##   0   0
##   2   0
##   3   0
##   4   0
##   5   0
##   6   0
##   7   0
##   8   0
##   9   0
##   10  0
##   11  0
##   12  0
##   13  0
##   14  0
##   15  0
##   16  0
##   17  0
##   18  0
library(ggplot2)

Initialise a plot of high_use and absences.

g3 <- ggplot(alc, aes(x = high_use, y = absences, col = sex))

Define the plot as a boxplot and draw it.

g3 + geom_boxplot() + ggtitle("Student absences by alcohol consumption")

For male students, high users of alcohol have more absences from school. For female students, the number of absences is quite similar weather they use high amount of alcohol or not.

Hypothesis 3: There is a relationship between alcohol use and student’s health. Initialise a plot of high_use and health.

g4 <- ggplot(alc, aes(x = high_use, y = health, col = sex))

Define the plot as a boxplot and draw it.

g4 + geom_boxplot() + ggtitle("Student health by alcohol consumption")

For male students, their health score was similar weather they were high users of alcohol or not. For female students, the health score was higher for those students who were high users of alcohol, surprisingly. In high alcohol users there was a lot more variation though.

Hypothesis 4: There is a relationship between alcohol use and student’s quality of family relationships. Initialise a plot of high_use and health.

g4 <- ggplot(alc, aes(x = high_use, y = goout, col = sex))

Define the plot as a boxplot and draw it.

g4 + geom_boxplot() + ggtitle("Going out with friends by alcohol consumption")

Both male and female high alcohol users go out with friends more than low alcohol users.

Find the model with glm().

m <- glm(high_use ~ absences + G3 + health + goout, data = alc, family = "binomial")

Print out a summary of the model.

summary(m)
## 
## Call:
## glm(formula = high_use ~ absences + G3 + health + goout, family = "binomial", 
##     data = alc)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.8599  -0.7560  -0.5555   0.9414   2.3160  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -3.60300    0.77329  -4.659 3.17e-06 ***
## absences     0.07508    0.02197   3.417 0.000633 ***
## G3          -0.04081    0.03843  -1.062 0.288295    
## health       0.12650    0.09087   1.392 0.163897    
## goout        0.72684    0.11849   6.134 8.57e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 465.68  on 381  degrees of freedom
## Residual deviance: 398.95  on 377  degrees of freedom
## AIC: 408.95
## 
## Number of Fisher Scoring iterations: 4

Print out the coefficients of the model.

coef(m)
## (Intercept)    absences          G3      health       goout 
## -3.60300340  0.07507726 -0.04080848  0.12650463  0.72684009

Absences and going out with friends are highly significant predictor of the probability of being a high user of alcohol. Health and grade are not significant predictor of the probability of being a high user of alcohol.

Present and interpret the coefficients of the model as odds ratios and provide confidence intervals for them. Find the model with glm().

m <- glm(high_use ~ absences + G3 + health + goout, data = alc, family = "binomial")

Compute odds ratios (OR).

OR <- coef(m) %>% exp

Compute confidence intervals (CI).

CI <- confint(m) %>% exp
## Waiting for profiling to be done...

Print out the odds ratios with their confidence intervals

cbind(OR, CI)
##                     OR       2.5 %   97.5 %
## (Intercept) 0.02724178 0.005694834 0.118851
## absences    1.07796743 1.034131788 1.128425
## G3          0.96001298 0.890279078 1.035492
## health      1.13485470 0.951360927 1.359672
## goout       2.06853388 1.649082084 2.626747

Absences, health, and going out with friends are all positively associated with high use of alcohol whereas grades is negatively associated with high alcohol use. Every absence causes the student 8 % more likely to be a high user of alcohol. For every increase in the health score the student is 13.5% more likely to be a high user of alcohol. Each time the student goes out with friends, they are 107 % more likely to be a high user of alcohol. For every increase in the grade the student is 4 % less likely to be a high user of alcohol.

Fit the model.

m <- glm(high_use ~ absences + goout, data = alc, family = "binomial")

Exploring the predictive power of the model.

Fit the model using only the variables that had a statistical relationship with high/low alcohol consumption.

m <- glm(high_use ~ absences + goout, data = alc, family = "binomial")

Predict() the probability of high_use.

probabilities <- predict(m, type = "response")

Add the predicted probabilities to ‘alc’.

alc <- mutate(alc, probability = probabilities)

Use the probabilities to make a prediction of high_use.

alc <- mutate(alc, prediction = probability > 0.5)

See the last ten original classes, predicted probabilities, and class predictions.

select(alc, failures, absences, sex, high_use, probability, prediction) %>% tail(10)
##     failures absences sex high_use probability prediction
## 373        1        0   M    FALSE  0.10204808      FALSE
## 374        1        7   M     TRUE  0.28913349      FALSE
## 375        0        1   F    FALSE  0.20395640      FALSE
## 376        0        6   F    FALSE  0.27356268      FALSE
## 377        1        2   F    FALSE  0.11705469      FALSE
## 378        0        2   F    FALSE  0.36613754      FALSE
## 379        2        2   F    FALSE  0.11705469      FALSE
## 380        0        3   F    FALSE  0.06419413      FALSE
## 381        0        4   M     TRUE  0.58446429       TRUE
## 382        0        2   M     TRUE  0.05971939      FALSE

Tabulate the target variable versus the predictions

table(high_use = alc$high_use, prediction = alc$prediction)
##         prediction
## high_use FALSE TRUE
##    FALSE   245   23
##    TRUE     67   47

23 times the prediction is “high alcohol use” when the variable is not “high alcohol use”. 67 times the prediction is not “high alcohol use” when the variable is “high alcohol use”.

Access dplyr and ggplot2.

library(dplyr); library(ggplot2)

A graphic visualizing both the actual values and the predictions.

Initialize a plot of ‘high_use’ versus ‘probability’ in ‘alc’.

g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))

Define the geom as points and draw the plot.

g + geom_point()

Tabulate the target variable versus the predictions.

table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table %>% addmargins
##         prediction
## high_use      FALSE       TRUE        Sum
##    FALSE 0.64136126 0.06020942 0.70157068
##    TRUE  0.17539267 0.12303665 0.29842932
##    Sum   0.81675393 0.18324607 1.00000000

According to the prediction, 82% of all the students are not high alcohol users. According to the actual values 70% of all the students are not high alcohol users. There is a quite big difference between the prediction and the actual model. Compute the total proportion of inaccurately classified individuals (the training error).

Define a loss function (average prediction error).

loss_func <- function(class, prob) {
  n_wrong <- abs(class - prob) > 0.5
  mean(n_wrong)
}

Call loss_func to compute the average number of wrong predictions in the (training) data.

loss_func(class = alc$high_use, prob = alc$probability)
## [1] 0.2356021

The training error is about 24%, which shows that the accuracy of the model is about 76%.


Exercise 4 Clustering and classification

#Access the MASS package.
library(MASS)
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
## 
##     select
#Load the data.
data("Boston")
#Explore the dataset.
str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08205   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
#Plot matrix of the variables.
pairs(Boston)

There are 506 observations (rows) and 14 variables (columns) in the dataset. The variables are for example crim (per capita crime rate by town), zn (proportion of residential land zoned for lots over 25,000 sq.ft), indus (proportion of non-retail business acres per town), chas (Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)), nox (nitrogen oxides concentration (parts per 10 million)), lstat (lower status of the population (percent)) and medv (median value of owner-occupied homes in $1000s).

For more details of the dataset, please visit https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/Boston.html

library(dplyr)
library(corrplot)
## corrplot 0.84 loaded
#Calculate the correlation matrix and round it.
cor_matrix<-cor(Boston) %>% round(digits = 2)
#Print the correlation matrix.
cor_matrix
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax ptratio
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58    0.29
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31   -0.39
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72    0.38
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04   -0.12
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67    0.19
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29   -0.36
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51    0.26
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53   -0.23
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91    0.46
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00    0.46
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46    1.00
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44   -0.18
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54    0.37
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47   -0.51
##         black lstat  medv
## crim    -0.39  0.46 -0.39
## zn       0.18 -0.41  0.36
## indus   -0.36  0.60 -0.48
## chas     0.05 -0.05  0.18
## nox     -0.38  0.59 -0.43
## rm       0.13 -0.61  0.70
## age     -0.27  0.60 -0.38
## dis      0.29 -0.50  0.25
## rad     -0.44  0.49 -0.38
## tax     -0.44  0.54 -0.47
## ptratio -0.18  0.37 -0.51
## black    1.00 -0.37  0.33
## lstat   -0.37  1.00 -0.74
## medv     0.33 -0.74  1.00
#Visualize the correlation matrix.
corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex = 0.6)

Positive correlations are displayed in blue and negative correlations in red color. Color intensity and the size of the circle are proportional to the correlation coefficients. There is a high negative correlation between indus (proportion of non-retail business acres per town) and dis (weighted mean of distances to five Boston employment centres), nox (nitrogen oxides concentration (parts per 10 million)) and dis (weighted mean of distances to five Boston employment centres), age (proportion of owner-occupied units built prior to 1940) and dis (weighted mean of distances to five Boston employment centres) and istat (lower status of the population (percent)) and medv (median value of owner-occupied homes in $1000s). There is a high positive correlation between rad (index of accessibility to radial highways) and tax (full-value property-tax rate per $10,000).

Standardize the dataset, create a categorical variable of the crime rate, divide the dataset to train and test sets.

#Center and standardize variables.
boston_scaled <- scale(Boston)
#Summaries of the scaled variables.
summary(boston_scaled)
##       crim                 zn               indus              chas        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109   Median :-0.2723  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648  
##       nox                rm               age               dis         
##  Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658  
##  1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049  
##  Median :-0.1441   Median :-0.1084   Median : 0.3171   Median :-0.2790  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617  
##  Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566  
##       rad               tax             ptratio            black        
##  Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033  
##  1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049  
##  Median :-0.5225   Median :-0.4642   Median : 0.2746   Median : 0.3808  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332  
##  Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406  
##      lstat              medv        
##  Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 3.5453   Max.   : 2.9865
#Class of the boston_scaled object.
class(boston_scaled)
## [1] "matrix" "array"
#Change the object to data frame.
boston_scaled <- as.data.frame(boston_scaled)

After scaling the mean is 0 for all the variables which means that all variables are normally distributed.

#Summary of the scaled crime rate.
summary(boston_scaled$crim)
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -0.419367 -0.410563 -0.390280  0.000000  0.007389  9.924110
#Create a quantile vector of crim and print it.
bins <- quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
#Create a categorical variable 'crime'.
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))
#Look at the table of the new factor crime.
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127
#Remove original crim from the dataset.
boston_scaled <- dplyr::select(boston_scaled, -crim)
#Add the new categorical value to scaled data.
boston_scaled <- data.frame(boston_scaled, crime)

Divide the dataset to train and test sets, so that 80% of the data belongs to the train set.

#Number of rows in the Boston dataset. 
n <- nrow(boston_scaled)
#Choose randomly 80% of the rows.
ind <- sample(n,  size = n * 0.8)
#Create train set.
train <- boston_scaled[ind,]
#Create test set.
test <- boston_scaled[-ind,]
#Save the correct classes from test data.
correct_classes <- test$crime
#Remove the crime variable from test data.
test <- dplyr::select(test, -crime)

Fit the linear discriminant analysis on the train set.

#Linear discriminant analysis.
lda.fit <- lda(crime ~ ., data = train)
#Print the lda.fit object.
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2673267 0.2475248 0.2351485 0.2500000 
## 
## Group means:
##                  zn      indus          chas        nox          rm        age
## low       0.8772850 -0.9134371 -0.1265105404 -0.8624409  0.40699008 -0.8726717
## med_low  -0.1373628 -0.2534824 -0.0361030535 -0.5597805 -0.14383991 -0.3390866
## med_high -0.3951672  0.2011651  0.1421025361  0.3868843  0.04443137  0.3933914
## high     -0.4872402  1.0171306  0.0005392655  1.0463235 -0.42841344  0.7755216
##                 dis        rad        tax     ptratio       black       lstat
## low       0.8324182 -0.6915643 -0.7490719 -0.37764028  0.38040704 -0.75794161
## med_low   0.3206544 -0.5466022 -0.4638572 -0.01179411  0.32268046 -0.13514283
## med_high -0.3662360 -0.4015931 -0.2946426 -0.25076937  0.07867359  0.02962702
## high     -0.8322378  1.6379981  1.5139626  0.78062517 -0.73516733  0.83703701
##                 medv
## low       0.50575191
## med_low  -0.03738202
## med_high  0.14373334
## high     -0.63430797
## 
## Coefficients of linear discriminants:
##                  LD1           LD2         LD3
## zn       0.066406232  0.7121604183 -0.83882403
## indus    0.069559539 -0.3278316938  0.38382153
## chas    -0.072867113 -0.0475829503  0.17379618
## nox      0.379091605 -0.5733555010 -1.48613799
## rm      -0.092708824 -0.0725724859 -0.08688787
## age      0.215247554 -0.3035426418 -0.20744471
## dis     -0.004234042 -0.2250641512  0.03326101
## rad      3.133885046  0.9768070413 -0.11926948
## tax      0.029945354 -0.0796940928  0.61905719
## ptratio  0.102999718  0.0741117683 -0.23945703
## black   -0.110366832  0.0001270964  0.12634402
## lstat    0.269135937 -0.2843819617  0.30771130
## medv     0.232173204 -0.3654241204 -0.35271066
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9507 0.0356 0.0137
#The function for lda biplot arrows.
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "orange", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}
#Target classes as numeric.
classes <- as.numeric(train$crime)
#Plot the lda results.
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)

The best linear separator is variable index of accessibility to radial highways (rad), which has the longest arrow in the picture. The second best separator is difficult to make out, but it looks like it would be nitrogen oxides concentration (parts per 10 million) (nox).

Predicting with the model.

#Predict classes with test data.
lda.pred <- predict(lda.fit, newdata = test)

#Cross tabulate the results.
table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       15       4        0    0
##   med_low    7      15        4    0
##   med_high   0      10       20    1
##   high       0       0        0   26

Note that the numbers change every time you run the model. The model predicted the high crime rates well (32/32 were classified correctly). Other categories the model did not predict as well: For medium high crime rates, 17/29 were classified correctly, for medium low crime rates 17/25 were classified correctly, and for low crime rates 9/16 were classified correctly.

total <- c(9+5+2+3+17+5+10+17+2+32)
total
## [1] 102
correct <- c(9+17+17+32)
correct
## [1] 75

Out of a total of 102 observations, 75 observations were classified correctly.

ratio <- c(correct/total)
ratio
## [1] 0.7352941

Accuracy of the model was 74%, which is not that bad but could be better.

Reload Boston dataset.

library(MASS)
data("Boston")
#Center and standardize variables.
boston_scaled <- scale(Boston)
#Change the object to data frame from matrix type.
boston_scaled <- as.data.frame(boston_scaled)
#Calculate the Euclidean distances between observations.
dist_eu <- dist(boston_scaled)
#Look at the summary of the distances.
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970

Run k-means algorithm on the dataset.

#K-means clustering.
km <-kmeans(boston_scaled, centers = 3)
#Plot the Boston dataset with clusters.
pairs(boston_scaled, col = km$cluster)

#Investigate the optimal number of clusters and run the algorithm again.
set.seed(123)
#Determine the number of clusters.
k_max <- 10
#Calculate the total within sum of squares.
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
#Visualize the results with qplot. Visualize (with qplot) the total WCSS when the number of cluster goes from 1 to 10.
library(ggplot2)
qplot(x = 1:k_max, y = twcss, geom = 'line')

2 clusters seems optimal as the bend (knee) is at 2.

#Run kmeans() again with two clusters.
km <-kmeans(boston_scaled, centers = 2)
#Plot the Boston dataset with clusters.
pairs(boston_scaled, col = km$cluster)

Like observed before, the optimal number of clusters seems to be two.

Super bonus.

model_predictors <- dplyr::select(train, -crime)
#Check the dimensions.
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
#Matrix multiplication.
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
#Matrix multiplication.
library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
#Create a 3D plot of the columns of the matrix product by typing the code below.
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')
## Warning: `arrange_()` is deprecated as of dplyr 0.7.0.
## Please use `arrange()` instead.
## See vignette('programming') for more help
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_warnings()` to see where this warning was generated.
#Add argument color as an argument in the plot_ly() function.
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)

Draw another 3D plot where the color is defined by the clusters of the k-means.

#Make a k-means with 4 clusters to compare the methods.
km3D <-kmeans(boston_scaled, centers = 4)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km3D$cluster[ind])

Medium high crime rates seems to bee better defined than cluster 1. Cluster 2 is a bit better defined (not so much intermingling) than low crime rate in the first picture. Cluster 3 is quite similar with high crime rates, even though cluster 3 is a bit more better defined. Cluster 4 is more defined than medium low crime rates in the first picture.


Exercise 5. Dimensionality reduction techniques

#Read the data and show summaries of the variables.
human<-read.table(file = "human.txt", sep="\t", header=TRUE)
str(human)
## 'data.frame':    155 obs. of  8 variables:
##  $ Edu2.FM  : num  1.007 0.997 0.983 0.989 0.969 ...
##  $ Labo.FM  : num  0.891 0.819 0.825 0.884 0.829 ...
##  $ Edu.Exp  : num  17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
##  $ Life.Exp : num  81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
##  $ GNI      : int  64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
##  $ Mat.Mor  : int  4 6 6 5 6 7 9 28 11 8 ...
##  $ Ado.Birth: num  7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
##  $ Parli.F  : num  39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...
summary(human)
##     Edu2.FM          Labo.FM          Edu.Exp         Life.Exp    
##  Min.   :0.1717   Min.   :0.1857   Min.   : 5.40   Min.   :49.00  
##  1st Qu.:0.7264   1st Qu.:0.5984   1st Qu.:11.25   1st Qu.:66.30  
##  Median :0.9375   Median :0.7535   Median :13.50   Median :74.20  
##  Mean   :0.8529   Mean   :0.7074   Mean   :13.18   Mean   :71.65  
##  3rd Qu.:0.9968   3rd Qu.:0.8535   3rd Qu.:15.20   3rd Qu.:77.25  
##  Max.   :1.4967   Max.   :1.0380   Max.   :20.20   Max.   :83.50  
##       GNI            Mat.Mor         Ado.Birth         Parli.F     
##  Min.   :   581   Min.   :   1.0   Min.   :  0.60   Min.   : 0.00  
##  1st Qu.:  4198   1st Qu.:  11.5   1st Qu.: 12.65   1st Qu.:12.40  
##  Median : 12040   Median :  49.0   Median : 33.60   Median :19.30  
##  Mean   : 17628   Mean   : 149.1   Mean   : 47.16   Mean   :20.91  
##  3rd Qu.: 24512   3rd Qu.: 190.0   3rd Qu.: 71.95   3rd Qu.:27.95  
##  Max.   :123124   Max.   :1100.0   Max.   :204.80   Max.   :57.50

There are 8 variables and 155 observations.

#Access GGally.
library(GGally)

#Visualize the variables.
ggpairs(human)

Many of the variables are skewed to either side and/or have long tails. Education expectancy looks the most normally distributed. Mean education expectancy for example is 13.2 years, mean life expectancy 71.7 years and mean adolescent birth rate 47.2. There is great variation within some variables: For the Gross national income per capita, the minimum value is 581, median 12040 and maximum 123124 and for the Maternal mortality ratio, the minimum value is 1, median 49 and maximum 1100. Then, in the Education expectancy variation is relatively small as the minimum value is 5.4, median 13.5 and maximum value 20.2.

There are several variables that are highly correlated with each other, for example: Adolescent birth rate and Maternal mortality, Life expectancy and Education expectancy, Life expectancy and Adolescent birth rate, Education expectancy and Maternal mortality.

Percentage of female representatives in parliament and Adolescent birth rate, for example, are not correlated.

Principal component analysis for non-standardized data

#Perform principal component analysis on the non-standardized data.
pca_human <- prcomp(human)
#Create and print out a summary of pca_human.
s <- summary(pca_human)
s
## Importance of components:
##                              PC1      PC2   PC3   PC4   PC5   PC6    PC7    PC8
## Standard deviation     1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912 0.1591
## Proportion of Variance 9.999e-01   0.0001  0.00  0.00 0.000 0.000 0.0000 0.0000
## Cumulative Proportion  9.999e-01   1.0000  1.00  1.00 1.000 1.000 1.0000 1.0000
# Rounded percetanges of variance captured by each PC.
pca_pr <- round(1*s$importance[2,]*100, digits = 1) 
#Print out the percentages of variance.
pca_pr
## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 
## 100   0   0   0   0   0   0   0
#Create object pc_lab to be used as axis labels.
pc_lab <- paste0(names(pca_pr), " (", pca_pr, "%)") 
#Draw a biplot.
biplot(pca_human, cex = c(0.8, 0.1), col = c("grey40", "deeppink2"), xlab = pc_lab[1], ylab = pc_lab[2])
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

Principal component analysis for standardized data

#Standardize the variables.
human_std <- scale(human)
#Print out summaries of the standardized variables.
summary(human_std)
##     Edu2.FM           Labo.FM           Edu.Exp           Life.Exp      
##  Min.   :-2.8189   Min.   :-2.6247   Min.   :-2.7378   Min.   :-2.7188  
##  1st Qu.:-0.5233   1st Qu.:-0.5484   1st Qu.:-0.6782   1st Qu.:-0.6425  
##  Median : 0.3503   Median : 0.2316   Median : 0.1140   Median : 0.3056  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5958   3rd Qu.: 0.7350   3rd Qu.: 0.7126   3rd Qu.: 0.6717  
##  Max.   : 2.6646   Max.   : 1.6632   Max.   : 2.4730   Max.   : 1.4218  
##       GNI             Mat.Mor          Ado.Birth          Parli.F       
##  Min.   :-0.9193   Min.   :-0.6992   Min.   :-1.1325   Min.   :-1.8203  
##  1st Qu.:-0.7243   1st Qu.:-0.6496   1st Qu.:-0.8394   1st Qu.:-0.7409  
##  Median :-0.3013   Median :-0.4726   Median :-0.3298   Median :-0.1403  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.3712   3rd Qu.: 0.1932   3rd Qu.: 0.6030   3rd Qu.: 0.6127  
##  Max.   : 5.6890   Max.   : 4.4899   Max.   : 3.8344   Max.   : 3.1850
#Perform principal component analysis (with the SVD method).
pca_human <- prcomp(human_std)
#Create and print out a summary of pca_human.
s <- summary(pca_human)
s
## Importance of components:
##                           PC1    PC2     PC3     PC4     PC5     PC6     PC7
## Standard deviation     2.0708 1.1397 0.87505 0.77886 0.66196 0.53631 0.45900
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595 0.02634
## Cumulative Proportion  0.5361 0.6984 0.79413 0.86996 0.92473 0.96069 0.98702
##                            PC8
## Standard deviation     0.32224
## Proportion of Variance 0.01298
## Cumulative Proportion  1.00000
#Rounded percetanges of variance captured by each PC.
pca_pr <- round(1*s$importance[2,]*100, digits = 1)
#Print out the percentages of variance.
pca_pr
##  PC1  PC2  PC3  PC4  PC5  PC6  PC7  PC8 
## 53.6 16.2  9.6  7.6  5.5  3.6  2.6  1.3
#Create object pc_lab to be used as axis labels.
pc_lab <- paste0(names(pca_pr), " (", pca_pr, "%)") 
#Draw a biplot.
biplot(pca_human, choices = 1:2, cex = c(0.6, 1), col = c("grey40", "deeppink2"), xlab = pc_lab[1], ylab = pc_lab[2], main = "Global health and equality related indicators") 

In the non-standardized data, the first principal component PC1 explains 100% of the variance and the second principal component PC2 explains 0% of the variance. In the standardized data, PC1 explains 53.6% of the variance and PC2 explains 16.2% of the variance. Together they explain 69.8% of the variance of the whole dataset. In PCA, the principal components are constructed so, that the first principal component accounts for the largest possible variance in the data set and the second one the second largest possible variance.

In the non-standardized data, there were variables with high range (great variance), which dominated the ones with smaller range, which lead to biased results. Standardization standardizes the variables so that every one of them contributes equally to the analysis, which prevents the problem of variables with larger range dominating and causing biased results.

In the non-standardized data, for example Gross national income GNI had such a big variance that it caused biased results. After standardization and removing the domination problem, the effects of the variables with smaller variation, for example education expectancy, can be seen.

Personal interpretations of the first two principal component dimensions

Iceland, Sweden, Norway, Finland and Denmark are closely grouped together and are therefore similar to each other. They all have quite high Percentage of female representatives in parliament, Expected years of education, Gross national income per capita, Population with secondary Education female to male ratio and life expectancy at birth.

The arrows visualize the connections between the original variables and the principal components. Expected years of education, Gross national income per capita, Population with secondary education female to male ratio and Life expectancy at birth have a small angle between them, so we can assume correlation between them. Maternal mortality ratio and adolescent birth rate also seem to be correlated. The angle of the arrows for Percentage of female representatives in parliament and Adolescent birth rate for example is quite large and therefore we can assume that the is no correlation between those variables.

Expected years of education, Gross national income per capita, Population with secondary education female to male ratio, Life expectancy at birth, Maternal mortality ratio and Adolescent birth rate all have a small angle with the principal component 1 and can be assumed to correlate with each other. Percentage of female representatives in parliament and Labour force participation rate female/male ratio on the other hand, can be assumed to correlate with principal component 2.

Sierra Leone, Liberia, Congo and Chad for example seem to have both high Maternal mortality ratio and Adolescent birth rate and do not have high Expected years of education, Gross national income per capita or life expectancy at birth. Mosambique has both quite high Maternal mortality ratio and Adolescent birth rate but women also have a high Labour force participation rate.

Tea dataset and multiple correspondence analysis

#Load library and tea dataset.
library(FactoMineR)
data(tea)
#Explore tea dataset.
dim(tea)
## [1] 300  36
str(tea)
## 'data.frame':    300 obs. of  36 variables:
##  $ breakfast       : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
##  $ tea.time        : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
##  $ evening         : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
##  $ lunch           : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dinner          : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
##  $ always          : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
##  $ home            : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
##  $ work            : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
##  $ tearoom         : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ friends         : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
##  $ resto           : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
##  $ pub             : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
##  $ Tea             : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How             : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ sugar           : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ how             : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ where           : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ price           : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
##  $ age             : int  39 45 47 23 48 21 37 36 40 37 ...
##  $ sex             : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
##  $ SPC             : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
##  $ Sport           : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
##  $ age_Q           : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
##  $ frequency       : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
##  $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
##  $ spirituality    : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
##  $ healthy         : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
##  $ diuretic        : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
##  $ friendliness    : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
##  $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
##  $ feminine        : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
##  $ sophisticated   : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
##  $ slimming        : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ exciting        : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
##  $ relaxing        : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
##  $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...
summary(tea)
##          breakfast           tea.time          evening          lunch    
##  breakfast    :144   Not.tea time:131   evening    :103   lunch    : 44  
##  Not.breakfast:156   tea time    :169   Not.evening:197   Not.lunch:256  
##                                                                          
##                                                                          
##                                                                          
##                                                                          
##                                                                          
##         dinner           always          home           work    
##  dinner    : 21   always    :103   home    :291   Not.work:213  
##  Not.dinner:279   Not.always:197   Not.home:  9   work    : 87  
##                                                                 
##                                                                 
##                                                                 
##                                                                 
##                                                                 
##         tearoom           friends          resto          pub     
##  Not.tearoom:242   friends    :196   Not.resto:221   Not.pub:237  
##  tearoom    : 58   Not.friends:104   resto    : 79   pub    : 63  
##                                                                   
##                                                                   
##                                                                   
##                                                                   
##                                                                   
##         Tea         How           sugar                     how     
##  black    : 74   alone:195   No.sugar:155   tea bag           :170  
##  Earl Grey:193   lemon: 33   sugar   :145   tea bag+unpackaged: 94  
##  green    : 33   milk : 63                  unpackaged        : 36  
##                  other:  9                                          
##                                                                     
##                                                                     
##                                                                     
##                   where                 price          age        sex    
##  chain store         :192   p_branded      : 95   Min.   :15.00   F:178  
##  chain store+tea shop: 78   p_cheap        :  7   1st Qu.:23.00   M:122  
##  tea shop            : 30   p_private label: 21   Median :32.00          
##                             p_unknown      : 12   Mean   :37.05          
##                             p_upscale      : 53   3rd Qu.:48.00          
##                             p_variable     :112   Max.   :90.00          
##                                                                          
##            SPC               Sport       age_Q          frequency  
##  employee    :59   Not.sportsman:121   15-24:92   1/day      : 95  
##  middle      :40   sportsman    :179   25-34:69   1 to 2/week: 44  
##  non-worker  :64                       35-44:40   +2/day     :127  
##  other worker:20                       45-59:61   3 to 6/week: 34  
##  senior      :35                       +60  :38                    
##  student     :70                                                   
##  workman     :12                                                   
##              escape.exoticism           spirituality        healthy   
##  escape-exoticism    :142     Not.spirituality:206   healthy    :210  
##  Not.escape-exoticism:158     spirituality    : 94   Not.healthy: 90  
##                                                                       
##                                                                       
##                                                                       
##                                                                       
##                                                                       
##          diuretic             friendliness            iron.absorption
##  diuretic    :174   friendliness    :242   iron absorption    : 31   
##  Not.diuretic:126   Not.friendliness: 58   Not.iron absorption:269   
##                                                                      
##                                                                      
##                                                                      
##                                                                      
##                                                                      
##          feminine             sophisticated        slimming          exciting  
##  feminine    :129   Not.sophisticated: 85   No.slimming:255   exciting   :116  
##  Not.feminine:171   sophisticated    :215   slimming   : 45   No.exciting:184  
##                                                                                
##                                                                                
##                                                                                
##                                                                                
##                                                                                
##         relaxing              effect.on.health
##  No.relaxing:113   effect on health   : 66    
##  relaxing   :187   No.effect on health:234    
##                                               
##                                               
##                                               
##                                               
## 

There are 36 variables and 300 observations. All the variables are factorial, except age, which is integral.

#Access library dplyr.
library(dplyr)
library(tidyr)

#Column names to keep in the dataset.
keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")

#Select the 'keep_columns' to create a new dataset.
tea_time <- dplyr::select(tea, one_of(keep_columns))

#Look at the summaries and structure of the data.
str(tea_time)
## 'data.frame':    300 obs. of  6 variables:
##  $ Tea  : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How  : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ how  : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
summary(tea)
##          breakfast           tea.time          evening          lunch    
##  breakfast    :144   Not.tea time:131   evening    :103   lunch    : 44  
##  Not.breakfast:156   tea time    :169   Not.evening:197   Not.lunch:256  
##                                                                          
##                                                                          
##                                                                          
##                                                                          
##                                                                          
##         dinner           always          home           work    
##  dinner    : 21   always    :103   home    :291   Not.work:213  
##  Not.dinner:279   Not.always:197   Not.home:  9   work    : 87  
##                                                                 
##                                                                 
##                                                                 
##                                                                 
##                                                                 
##         tearoom           friends          resto          pub     
##  Not.tearoom:242   friends    :196   Not.resto:221   Not.pub:237  
##  tearoom    : 58   Not.friends:104   resto    : 79   pub    : 63  
##                                                                   
##                                                                   
##                                                                   
##                                                                   
##                                                                   
##         Tea         How           sugar                     how     
##  black    : 74   alone:195   No.sugar:155   tea bag           :170  
##  Earl Grey:193   lemon: 33   sugar   :145   tea bag+unpackaged: 94  
##  green    : 33   milk : 63                  unpackaged        : 36  
##                  other:  9                                          
##                                                                     
##                                                                     
##                                                                     
##                   where                 price          age        sex    
##  chain store         :192   p_branded      : 95   Min.   :15.00   F:178  
##  chain store+tea shop: 78   p_cheap        :  7   1st Qu.:23.00   M:122  
##  tea shop            : 30   p_private label: 21   Median :32.00          
##                             p_unknown      : 12   Mean   :37.05          
##                             p_upscale      : 53   3rd Qu.:48.00          
##                             p_variable     :112   Max.   :90.00          
##                                                                          
##            SPC               Sport       age_Q          frequency  
##  employee    :59   Not.sportsman:121   15-24:92   1/day      : 95  
##  middle      :40   sportsman    :179   25-34:69   1 to 2/week: 44  
##  non-worker  :64                       35-44:40   +2/day     :127  
##  other worker:20                       45-59:61   3 to 6/week: 34  
##  senior      :35                       +60  :38                    
##  student     :70                                                   
##  workman     :12                                                   
##              escape.exoticism           spirituality        healthy   
##  escape-exoticism    :142     Not.spirituality:206   healthy    :210  
##  Not.escape-exoticism:158     spirituality    : 94   Not.healthy: 90  
##                                                                       
##                                                                       
##                                                                       
##                                                                       
##                                                                       
##          diuretic             friendliness            iron.absorption
##  diuretic    :174   friendliness    :242   iron absorption    : 31   
##  Not.diuretic:126   Not.friendliness: 58   Not.iron absorption:269   
##                                                                      
##                                                                      
##                                                                      
##                                                                      
##                                                                      
##          feminine             sophisticated        slimming          exciting  
##  feminine    :129   Not.sophisticated: 85   No.slimming:255   exciting   :116  
##  Not.feminine:171   sophisticated    :215   slimming   : 45   No.exciting:184  
##                                                                                
##                                                                                
##                                                                                
##                                                                                
##                                                                                
##         relaxing              effect.on.health
##  No.relaxing:113   effect on health   : 66    
##  relaxing   :187   No.effect on health:234    
##                                               
##                                               
##                                               
##                                               
## 

There are now 6 variables and 300 observations.

#Multiple correspondence analysis.
mca <- MCA(tea_time, graph = FALSE)

#Summary of the model.
summary(mca)
## 
## Call:
## MCA(X = tea_time, graph = FALSE) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6   Dim.7
## Variance               0.279   0.261   0.219   0.189   0.177   0.156   0.144
## % of var.             15.238  14.232  11.964  10.333   9.667   8.519   7.841
## Cumulative % of var.  15.238  29.471  41.435  51.768  61.434  69.953  77.794
##                        Dim.8   Dim.9  Dim.10  Dim.11
## Variance               0.141   0.117   0.087   0.062
## % of var.              7.705   6.392   4.724   3.385
## Cumulative % of var.  85.500  91.891  96.615 100.000
## 
## Individuals (the 10 first)
##                       Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## 1                  | -0.298  0.106  0.086 | -0.328  0.137  0.105 | -0.327
## 2                  | -0.237  0.067  0.036 | -0.136  0.024  0.012 | -0.695
## 3                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 4                  | -0.530  0.335  0.460 | -0.318  0.129  0.166 |  0.211
## 5                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 6                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 7                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 8                  | -0.237  0.067  0.036 | -0.136  0.024  0.012 | -0.695
## 9                  |  0.143  0.024  0.012 |  0.871  0.969  0.435 | -0.067
## 10                 |  0.476  0.271  0.140 |  0.687  0.604  0.291 | -0.650
##                       ctr   cos2  
## 1                   0.163  0.104 |
## 2                   0.735  0.314 |
## 3                   0.062  0.069 |
## 4                   0.068  0.073 |
## 5                   0.062  0.069 |
## 6                   0.062  0.069 |
## 7                   0.062  0.069 |
## 8                   0.735  0.314 |
## 9                   0.007  0.003 |
## 10                  0.643  0.261 |
## 
## Categories (the 10 first)
##                        Dim.1     ctr    cos2  v.test     Dim.2     ctr    cos2
## black              |   0.473   3.288   0.073   4.677 |   0.094   0.139   0.003
## Earl Grey          |  -0.264   2.680   0.126  -6.137 |   0.123   0.626   0.027
## green              |   0.486   1.547   0.029   2.952 |  -0.933   6.111   0.107
## alone              |  -0.018   0.012   0.001  -0.418 |  -0.262   2.841   0.127
## lemon              |   0.669   2.938   0.055   4.068 |   0.531   1.979   0.035
## milk               |  -0.337   1.420   0.030  -3.002 |   0.272   0.990   0.020
## other              |   0.288   0.148   0.003   0.876 |   1.820   6.347   0.102
## tea bag            |  -0.608  12.499   0.483 -12.023 |  -0.351   4.459   0.161
## tea bag+unpackaged |   0.350   2.289   0.056   4.088 |   1.024  20.968   0.478
## unpackaged         |   1.958  27.432   0.523  12.499 |  -1.015   7.898   0.141
##                     v.test     Dim.3     ctr    cos2  v.test  
## black                0.929 |  -1.081  21.888   0.382 -10.692 |
## Earl Grey            2.867 |   0.433   9.160   0.338  10.053 |
## green               -5.669 |  -0.108   0.098   0.001  -0.659 |
## alone               -6.164 |  -0.113   0.627   0.024  -2.655 |
## lemon                3.226 |   1.329  14.771   0.218   8.081 |
## milk                 2.422 |   0.013   0.003   0.000   0.116 |
## other                5.534 |  -2.524  14.526   0.197  -7.676 |
## tea bag             -6.941 |  -0.065   0.183   0.006  -1.287 |
## tea bag+unpackaged  11.956 |   0.019   0.009   0.000   0.226 |
## unpackaged          -6.482 |   0.257   0.602   0.009   1.640 |
## 
## Categorical variables (eta2)
##                      Dim.1 Dim.2 Dim.3  
## Tea                | 0.126 0.108 0.410 |
## How                | 0.076 0.190 0.394 |
## how                | 0.708 0.522 0.010 |
## sugar              | 0.065 0.001 0.336 |
## where              | 0.702 0.681 0.055 |
## lunch              | 0.000 0.064 0.111 |
#Visualize the dataset.
gather(tea_time) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
## Warning: attributes are not identical across measure variables;
## they will be dropped

#Visualize Multiple Correspondence Analysis.
plot(mca, invisible=c("ind"), habillage="quali")

People who drink unpackaged tea buy it from tea shops and people who use tea bags for theit tea buy it from the chain stores. People who drink both unpackaged tea and use tea bags also buy their tea from both types of shops.